Psalm 51:7-9 meaning

David now turns to God with the cry of the heart. He has recognized the need for truth to implant in his innermost being (Psalm 51:6). But now he recognizes that it is only God that can clean out the depth of his own sin; he can't do it himself. David cries out to God to Purify me with hyssop, and I shall be clean; Wash me and I shall be whiter than snow (v 7). 

The use of hyssop was in reference to a purification rite that would see the hyssop plant, or some similar aromatic plant, be dipped in water and sprinkled in order to purify. God commanded the children of Israel to use hyssop to dip in blood and to sprinkle the blood on the doorposts for the Passover (Exodus 12:22). The blood caused the death angel to pass over the house of those marked with blood. This looked forward to the blood of Christ, who is our passover. 

The word purify used here would indicate that David is asking for a purification, or cleansing that would free him from the adverse effects of sin. One of the adverse effects is the separation of fellowship between he and God. 

The picture here appears to be the purification rite with hyssop as an outward, visible sign that represents an inner cleansing of the heart. 

David has already asked God to wash him (v 2) and purify him (v7). He repeats now his request of God: Wash me, and I shall be whiter than snow. Here David expresses confidence that if God does the washing, the result will be certain; David will be made whiter than snow. Whiter than snow describes a crisp, pristine purity of heart and spirit, untouched by those things that corrupt the human soul and separate the creature from the Creator. 

David yearns for the rekindling of his broken relationship with God. That means God's righteousness must overtake and transform all the darkness of sin that had polluted the psalmist's life. The Law of Moses—delivered in the first five books of the Old Testament, the Torah—warned against and disclosed the depths of sin to which people could descend. God's grace alone is able effectively to wash—to clean, to make pure—and thereby reunite the fallen human spirit with the One who brings into being and sustains it (Ecclesiastes 12:7; Zechariah 12:1). The greatest gift of purifying, transformative grace God has extended to all humanity is in the person of His Son, Jesus.

David realizes he needs a complete cleansing, something that will penetrate to the very heart. 

New Testament believers need the same, for ongoing fellowship with God and with one another. Thankfully, through Jesus Christ, we can see how completely God has made provision for purification and cleansing of the heart. 

Cleansing of the heart begins with our initial salvation, when we are delivered once and for all from the penalty of sin, which is eternal separation from God and His family. New Testament believers receive the free gift of eternal life when they believe (John 3:14-15). 

This is confirmed by the writer of the book of Hebrews who writes, 

"When He [Jesus] had made purification of sins, He sat down at the right hand of the Majesty on high."
(Hebrews 1:3b)

The image of Jesus sitting down at the right hand of His Father invokes two realities: 

1. Jesus's purification for sins through His death on the cross was sufficient to cover all sins for all time (Colossians 2:14). That He sat down means that His work was finished. 
2. Because Jesus did His Father's will, even to death on a cross, He was elevated above all things (Philippians 2:9-10). 

When Peter testified to the his fellow apostles and other church leaders in Jerusalem about his ministry to the Gentiles, he verified of the work of God in the heart of the Gentiles by speaking in terms of purification, "And God, who knows the heart, testified to them giving them the Holy Spirit, just as He did to us; and He made no distinction between us and them, cleansing their hearts by faith" (Acts 15:8-9).

But David, being a man after God's heart, is like a New Testament believer who has already received the free gift of eternal salvation. Existing believers do not need cleansing from sin to be made righteous in God's sight. Existing believers need to be cleansed from sin to restore fellowship with God and with one another. 

The idea of confession being a key to restoring fellowship with God and man is a theme in John's first epistle: 

"but if we walk in the Light as He Himself is in the Light, we have fellowship with one another, and the blood of Jesus His Son cleanses us from all sin."
(1 John 1:7)

Thus, the New Testament moves from the rite of purification with hyssop (v 7) to the application of the blood of Christ for the ongoing purification and cleansing of the heart, and restoration of fellowship with God and with one another. 

David continues:

Make me to hear joy and gladness,
Let the bones which You have broken rejoice (v 8). 

Some of the effects, the negative consequences of sin that David is asking God to "wash," "cleanse," "blot out," and "purify" are mentioned here. He feels the crush of sin and guilt as he cries out Let the bones which You have broken rejoice. 

As Nathan prophesied, David's sins would be forgiven, but the consequences of his sin would live on with him: 

"Now therefore, the sword shall never depart from your house, because you have despised Me and have taken the wife of Uriah the Hittite to be your wife. Thus says the Lord, 'Behold, I will raise up evil against you from your own household; I will even take your wives before your eyes and give them to your companion, and he will lie with your wives in broad daylight.'"
(2 Samuel 12:10-11)

The Hebrew word translated bones in this instance represents David's entire being. God's chastisement of David has caused him to feel crushed, broken. Why would God do this? Because to leave David in sin is a much greater harm than God's discipline. As God says in the New Testament:

"Those whom I love, I reprove and discipline; therefore be zealous and repent."
(Revelation 3:19)

As painful as the consequences would be of David's sin and God's chastisement, David appears to be more concerned with what was happening inwardly in his "innermost being." He pleads with God to 

Hide Your face from my sins
And blot out all my iniquities (9). 

This is an extraordinary statement from David. Throughout the Old Testament, when there is a reference to God hiding His face, it would seem to indicate a broken fellowship. This is something the speaker desires to avoid; there is typically a plea for God not to hide His face. 

But here David is appealing to God to Hide Your face not from the person of David, not from fellowship with David, but from his sins. As he did in verse 1, he appeals again for God to blot out all his iniquities. The idea seems to be that this would be a preamble to God restoring fellowship with David. 

David knows he does not deserve this, but relies on God's grace, as he stated in verse 1,

"Be gracious to me, O God, according to your lovingkindness;
According to the greatness of Your compassion."

This is the only hope of appeal for David. It is the same for all of us. Thankfully we have a merciful God who desires to forgive (1 John 1:9).